Self-Consistent Model of Concentration and Temperature

The filled symbols indicate points corresponding to PDMS and have been included in determining the constants of the linear free energy relationship...
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Ind. Eng. Chem. Res. 2005, 44, 1547-1556

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Self-Consistent Model of Concentration and Temperature Dependence of Permeability in Rubbery Polymers Rajeev S. Prabhakar,†,§ Roy Raharjo,† Lora G. Toy,‡ Haiqing Lin,† and Benny D. Freeman*,† Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas 78758, and RTI International, Research Triangle Park, North Carolina 27709

A model describing the concentration and temperature dependences of gas and vapor permeability in rubbery polymers is presented. The solubility and permeability of propane in poly(dimethylsiloxane) (PDMS), a commercially used vapor separation membrane material, were determined over a wide range of temperatures and pressures to test the model. The model describes propane permeability in PDMS with an average error of 8.2%. The model also accurately predicts a decrease in propane permeability in PDMS with decreasing permeate pressure at fixed feed pressure. The model has also been tested satisfactorily using literature data for transport of condensable penetrants in PDMS and poly(ethylene). Introduction Membrane-based gas separation processes have gained acceptance in the chemical industry due to advantages of low cost, low energy requirements, and low maintenance relative to conventional technologies.1 Membranes are already used for separating mixtures of light gases: for example, in the production of high-purity nitrogen from air, removal of H2 from ammonia purge streams, H2-CH4 separation in refinery off-gases, H2/ CO adjustment in oxo-chemical synthesis plants, and CO2 separation from CH4 in natural gas processing.1 Membranes are also finding use in the separation of organic vapors from vent-gas streams in the chemical, petrochemical, and pharmaceutical industries.2 Applications such as hydrocarbon vapor separation from hydrogen in refineries and hydrocarbon dewpointing of natural gas are at the threshold of commercialization.2 A distinguishing feature of these newer applications is the separation of highly condensable penetrants from each other or from light gases. To optimize membrane system design, its performance is often evaluated under various operating conditions (i.e., temperature, feed pressure, permeate pressure, etc.), and this evaluation requires estimates of gas permeability coefficients over a wide range of conditions. Often, permeability data are not available over the complete range of conditions one would want to explore, possibly by computer simulation, in considering design alternatives. Therefore, it becomes necessary to estimate permeation properties based on extrapolation from known experimental data. The objective of this study has been to develop a rational framework to guide the estimation of permeability under conditions away from those where experimental data are available. The focus is on modeling the transport of gases and vapors in rubbery polymers since such materials are being used in applications where permeability is a strong function * To whom correspondence should be addressed. E-mail: [email protected]. Phone: (512) 232-2803. Fax: (512) 232-2807. † The University of Texas at Austin. § Present address: MEDAL/Air Liquide, Newport, DE 19804. ‡ RTI International.

of both temperature and upstream and downstream pressures. The model is based on fundamental and wellaccepted principles of gas solubility in rubbery polymers, activated Fickian diffusion of small molecules in polymers, and a judicious use of reasonable empirical approximations in cases where available theory does not provide a straightforward and simple element for the model. In all cases, pure gas permeability and sorption data are used to test the model because (i) no systematic experimental studies of mixed-gas permeation and sorption properties in rubbery polymers exist to permit reasonable validation of a model that included such effects and (ii) available experience from industrial sources suggests that, for many of the applications mentioned above, mixture effects have less impact on permeability than temperature and pressure. The following section describes the existing background information, presents the conventional framework for interpreting the temperature and pressure dependences of permeability, and illustrates the shortcomings of this approach when permeability is a strong function of penetrant concentration in the polymer. The basis for the new model is then presented, and the relevant equations are derived. Finally, the model is compared with experimental data. Background The permeability of permanent gases in polymers at a fixed temperature is very often constant at low to moderate pressures. The temperature dependence of the permeability coefficient over limited temperature ranges away from polymer thermal transitions can usually be described satisfactorily by the Arrhenius-type equation3

P ) Po exp

( ) -EP RT

(1)

where Po is the preexponential coefficient, EP is the activation energy of permeation, R is the universal gas constant, and T is absolute temperature. Thus, permeabilities measured over a range of temperatures at fixed feed and permeate pressures are sufficient to fit the above equation. This equation can then be used to

10.1021/ie0492909 CCC: $30.25 © 2005 American Chemical Society Published on Web 02/05/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005

Figure 1. Illustration of the graphical technique for using eq 1 to describe pressure- and temperature-dependent penetrant permeability in a polymer. The experimentally measured permeabilities are shown in part a. These data are replotted, at fixed feed pressures (p1, p2, p3, and p4), in part b to determine the adjustable parameters, EP and Po, from the slope and intercept, respectively, of the best-fit trend line through the data. The values of these parameters at different pressures are then plotted in parts c and d, respectively, to determine their pressure dependence. This graphical method requires at least 10 fitting parameters: 6 for part b and 2 each for parts c and d.

predict permeability coefficients at different temperatures and pressures that may be of interest for the design of a membrane separation system. For more condensable penetrants, which are increasingly being encountered in newer separations, the solubility and/or diffusion coefficients depend, often strongly, on penetrant concentration in the polymer.4-12 Therefore, the permeability coefficient also depends on penetrant concentration and, hence, on temperature as well as feed and permeate pressures. In such cases, permeability coefficient estimates are required at each combination of pressures and temperature being considered for the membrane system. The experimental determination of permeability coefficients under all operating conditions of interest can be a prohibitive task. Therefore, a theoretical framework that can guide the estimation of permeability coefficients as a function of these processing conditions using a limited amount of experimental data could be useful for designing membrane systems. Predicting the effects of pressure and temperature on permeability coefficients is often done by a simple extension of eq 1 (cf. parts a-d of Figure 1). Permeability coefficients are first determined experimentally over a range of feed pressures and temperatures, usually at fixed permeate pressure (Figure 1a). From these data, permeability values are obtained at fixed feed pressure (i.e., p1, p2, p3, and p4 in Figure 1a), often by interpolation, and over the entire temperature range.

These values are used to obtain Po and EP values by a least-squares fit of eq 1 (Figure 1b). When this process is repeated at different feed pressures, the values of Po and EP can be determined over a range of feed pressures (cf. Figure 1c,d). Equation 1 can then be used, along with the best-fit values of the adjustable constants, to estimate gas permeability at various combinations of feed pressure and temperature. The above methodology has several drawbacks. It requires many empirical adjustable constants to fit eq 1 using the graphical method outlined in Figure 1. Moreover, these fitting parameters may have little or no physical significance, which can sharply compromise the ability to estimate permeability coefficients beyond the temperature and pressure windows that have been explored experimentally. Finally, this methodology cannot account for the effect of changes in permeate pressure on the permeability coefficient. In other words, if a membrane is to be operated at a permeate pressure other than that for which experimental data are available and if permeability is sensitive to permeate pressure, then this method fails to provide a pathway for rational extrapolation of the data. Therefore, a method circumventing these disadvantages and providing predictions of the concentration and temperature dependences of gas and vapor permeability in polymers based on limited experimental data would be useful. As a first step in this direction, we describe a model that yields algebraic expressions to describe the effect of temperature and feed as well as permeate pressure on pure gas permeability coefficients. This model being proposed here requires very few fitting parameters. The utility of the model and its capabilities are demonstrated by correlating data from the literature and our laboratory for the transport of organic vapors in rubbery polymers. Poly(dimethylsiloxane) (PDMS) was chosen as the model polymer because it is used commercially as a vapor separation membrane material.1 The permeability and solubility of a condensable hydrocarbon, propane, were measured over a wide range of feed pressures and temperatures to obtain data for testing the model. Permeability measurements were also performed at two different permeate pressures to study the effect of permeate pressure changes on vapor permeability in this polymer. These experimental data were utilized to test the capability of the model for predicting the effect of change in permeate pressure on vapor permeability. The model was also tested with data from the literature for the transport of halothane, a commercial anesthetic, in PDMS and of several organic compounds in poly(ethylene) (PE). The main selection criterion for data was the availability of both solubility and permeability coefficients for the penetrant in the polymer as a function of pressure and temperature. Detailed information about the model cases is recorded in Table 1. Theory Small-molecule transport in polymer membranes is widely modeled using the solution-diffusion mechanism3 and is expressed by a permeability coefficient, P, defined as

P)

Nl p2 - p1

(2)

where N is the steady-state gas flux through a polymer membrane of thickness l due to a partial pressure

Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1549 Table 1. Solubility and Permeability Data Sources solubility data

permeability data

penetrant

polymer

temp range (°C)

no. of isotherms

total no. of points

temp range (°C)

no. of isotherms

total no. of points

ref

propane halothanea methyl bromide isobutylene n-hexane

PDMS PDMS PE PE PE

0 to 55 21 to 50 0 to 30 -8 to +30 0 to 30

4 6 2 3 2

65 35 11 8 10

-20 to +55 17 to 60 0 to 30 -8 to +30 0 to 30

5 5 2 3 2

45 27 8 8 7

this study 9 8 8 8

a

The chemical formula of halothane is CF3CHClBr.

difference (p2 - p1) across the film, p2 is the feed (upstream) pressure, and p1 is the permeate (downstream) pressure. In the simplest case, penetrant diffusion is modeled using Fick’s law of diffusion:13

Dloc dC N)1 - ω dx

( )

(3)

where Dloc is the local diffusion coefficient, C is the penetrant concentration, ω is the penetrant mass fraction in the polymer, and x is the direction of diffusion. Combining eqs 2 and 3 and integrating across the film thickness yield

P)

1 p2 - p1

∫CC D dC 2

1

(4)

where C2 and C1 are the penetrant concentrations at the upstream and downstream faces, respectively, of the polymer membrane at a given temperature and D is the effective diffusion coefficient in the polymer, defined for convenience as

D≡

Dloc 1-ω

(5)

If the diffusion coefficient is not a function of concentration, eq 4 reduces to

P)

C2 - C1 D p2 - p1

(6)

In the limit of negligible permeate pressure or if the sorption process obeys Henry’s law,3 eq 6 gives the familiar result

P ) SD

(7)

where the gas solubility coefficient S is given by C2/p2, the ratio of the gas concentration at the upstream face of the membrane to the upstream partial pressure of the gas. Equation 7 is often used to calculate gas flux in membrane-based separation processes. However, if the diffusion coefficient is a function of penetrant concentration and temperature, then an appropriate form of this function must be chosen and substituted into eq 4 to determine the permeability coefficient. The temperature dependence of the diffusion coefficient is typically well-described by the activated diffusion model of Arrhenius:14

D ) Do exp

( ) -ED RT

(8)

where Do is the preexponential coefficient and ED is the activation energy of diffusion. Interestingly, it has been observed that Do and ED in this equation are not

independent. Using diffusivity data for light gases in several rubbery polymers over the temperature range 17-50 °C, van Amerongen found that15

ED -b R

ln Do ) a

(9)

where a has a value of 0.0023 K-1 and b is 9.7 when Do has units of cm2/s. This equation is the “linear free energy relationship”. Barrer used a slightly different expression to describe the relationship between Do and ED:16

ED - b′ ln Do ) a′ RT

(10)

Barrer utilized van Amerongen’s data,15 as well as diffusivity values of permanent gases and light hydrocarbons over different temperature ranges in several rubbery polymers,16,17 to obtain best-fit values for a′ and b′. The reported values are 0.64 for a′ and 8.3 for b′ when Do has units of cm2/s. While eq 10 appears to be a more general form of eq 9, it leads to an apparent inconsistency. If eq 10 is substituted into eq 8, one obtains

D ) e-b′ exp

(

)

(a′ - 1)ED RT

(11)

Equation 11 implies that a plot of the natural logarithm of D versus 1/T should result in a fixed intercept of -b′, irrespective of the value of ED and, hence, of the polymer-gas system. This is contradictory to the linear free energy relationship and is obviously not the case.15-17 However, the equation proposed by van Amerongen does not suffer from the inconsistency issue, as can be seen by substituting eq 9 into eq 8:

D ) ea(ED/R)-b exp

( ) -ED RT

(12)

From eq 12, a plot of the natural logarithm of D versus 1/T results in an intercept which depends on ED, as described by the linear free energy relationship. Equation 9 is, therefore, rationalized to be the more consistent form of the empirical linear free energy relationship. Figure 2 presents the least-squares best-fit form of eq 9 to experimental data from several sources to demonstrate the correlation between Do and ED.15-21 These data include the original data utilized by van Amerongen and Barrer as well as more recent reports of Do and ED values for penetrant diffusion in rubbery polymers. When Do is expressed in units of cm2/s, eq 9 fits the data well with a and b values of 2.0 × 10-3 K-1 and 8.3, respectively (Figure 2), which are very similar

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vf(T,p,φ) ) vfs(Ts,ps,0) + R(T - Ts) - β(p - ps) + γφ (17)

Figure 2. Linear free energy relationship based on data for transport of permanent gases and hydrocarbons in several rubbery polymers.15-21 The least-squares best-fit line in the figure has the equation ln Do (cm2/s) ) (2.0 × 10-3 K-1)ED/R - 8.3. The filled symbols indicate points corresponding to PDMS and have been included in determining the constants of the linear free energy relationship.

to the values obtained by van Amerongen almost 60 years ago. These updated values of the fitting parameters are used in the proposed model for the temperature and concentration dependence of permeability. For the ensuing model discussion, the following more convenient form obtained by rearranging eq 12 is used:

D ) e-b exp(RED)

where vfs(Ts,ps,0) is the free volume of the pure polymer at some reference temperature, Ts, and pressure, ps, and R, β, and γ are adjustable constants. R and β are often set to the polymer’s thermal expansion coefficient and compressibility, respectively. This model (eqs 16 and 17) has many parameters that need to be determined by independent experiments or estimated by data-fitting techniques. Also, Rogers et al. have shown that, at low sorbed concentrations, the above model degenerates (via a Taylor series expansion) to the empirical model (eq 15).8 From a practical standpoint, for the development of the model in this study, eq 15 is advantageous due to its simplicity. From eqs 13 and 14, D is a function of ED and T. Therefore, the concentration dependence of the diffusion coefficient must arise from to a concentration dependence of ED. From a comparison of eqs 13 and 15, an exponential dependence of the diffusion coefficient on concentration requires a linear dependence of the activation energy on concentration. Thus, as a first approximation,

ED ) EDo(1 - kC)

(18)

where EDo is the activation energy of diffusion in the infinite dilution limit and k is an adjustable constant that describes the effect of penetrant concentration in the polymer on ED. From eqs 4, 13, 14, and 18

(13) P)

where

e-b eRED2 - eRED1 p2 - p1 -RE ok

(19)

D

R)

1 1 aR T

(

)

(14)

Concentration Dependence of the Diffusion Coefficient. Penetrant diffusion coefficients in rubbery polymers typically increase with increasing penetrant concentration in the polymer.4-12 In many cases, the following simple model is used to describe this concentration dependence:5,7,22,23

D ) eλC

(15)

where  and λ are adjustable constants and C is the penetrant concentration in the polymer. Some reports use penetrant volume fraction6,7 or penetrant activity,8 instead of concentration, in eq 15. Free volume theory is often invoked to explain this concentration dependence. In the free volume model, D is given by24

( )

D ) A exp -

B vf

(16)

where A and B are adjustable constants and vf is the volume fraction of the free volume of the polymer (called free volume, henceforth). The free volume of the polymer is often expressed as a function of three thermodynamic variables: temperature, T; hydrostatic pressure, p, applied to the penetrant-polymer system; and penetrant concentration in the polymer, which is usually expressed as the penetrant volume fraction in the polymer, φ:25

where

EDn ) EDo(1 - kCn)

n ) 1, 2

(20)

where subscripts 1 and 2 refer to the downstream and upstream faces of the membrane, respectively. Equation 19 incorporates the concentration dependence of the diffusion coefficient and also explicitly describes the effect of permeate pressure, p1, on the permeability coefficient. Equation 19 requires concentration values at the pressures and temperature at which the permeability coefficients are to be calculated. Therefore, an appropriate sorption model is needed to calculate these penetrant concentration values. The cases in this study correspond to condensable vapor sorption and transport in rubbery polymers. The uptake of condensable vapors in a rubbery polymer is frequently modeled using the Flory-Huggins expression:26

ln a ) ln φ2 + (1 - φ2) + χ(1 - φ2)2

(21)

where a is penetrant activity in the vapor phase, φ2 is the volume fraction of sorbed penetrant, and χ is the Flory-Huggins interaction parameter. For cross-linked rubbers, one may also consider the Flory-Rehner model,26 which accounts for the influence of cross-links on the penetrant free energy in the polymer. However, from a practical viewpoint, such effects are often quite small for the industrial examples mentioned earlier; therefore, we have used the simpler Flory-Huggins

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model in this development. In this study, penetrant activity is set equal to the relative pressure, p/psat, where psat is the saturation vapor pressure of the penetrant. The volume fraction of sorbed penetrant, φ2, is calculated from the equilibrium penetrant concentration in the polymer, C, as

[

φ2 ) 1 +

]

22,414 CV h2

-1

(22)

where V h 2 is the penetrant partial molar volume and is estimated as described previously.11 In this equation, C and V h 2 have units of cm3(STP)/cm3 polymer and cm3/ mol, respectively. The number 22,414 is a conversion factor with units of cm3(STP)/mol. The Flory-Huggins interaction parameter χ describes the interaction between penetrant and polymer and is a function of temperature, T.27 In some cases, χ also varies with penetrant concentration in the polymer.11 The concentration dependence is normally described adequately by a power series.28 Although many forms have been proposed to describe the temperature and concentration dependences of χ,28 in this work, these two dependencies have been empirically combined into a single equation as

χ ) χ0 +

χ1 + χ2(1 - φ2) T

(23)

where χ0, χ1, and χ2 are adjustable constants. The need for a temperature or concentration dependence of χ for a particular penetrant-polymer pair can be determined by performing an F test on the results obtained by fitting sorption data to eqs 21-23 with different numbers of adjustable constants in eq 23.29 Experimental Section Materials. PDMS films were prepared from an isooctane solution of 40 wt % Dehesive 940A silicone (Wacker Silicones Corporation, Adrian, MI). As supplied by the manufacturer, the Dehesive 940A silicone product is a viscous 30 wt % silicone gum in naphtha solvent. Before casting, the proprietary Crosslinker V24/Catalyst OL system provided by Wacker Silicones Corp. was added to the polymer solution. The films were made by pouring the polymer solution into a casting ring supported by a glass plate. The cast films were dried slowly under ambient conditions for 4 days. They were then placed in an oven at 110 °C for 30 min to remove residual solvent and to fully cross-link the polymer. After they were cooled to room temperature, the crosslinked films were easily removed from the casting ring and glass plate. Finally, the films were washed with n-heptane in a Soxhlet extractor for 3 days to remove impurities (i.e., unreacted cross-linker and catalyst). The resulting PDMS films were transparent and rubbery and were not tacky. Film thicknesses were determined with a digital micrometer readable to (1 µm and were 220 µm for the permeation samples. The density of the PDMS films was 0.99 g/cm3, and their cross-link density was approximately 4.93 × 10-4 mol/cm3. The cross-link density was measured as described previously.10 Chemical-grade propane of purity 99% was purchased from Matheson Tri-Gas (Austin, TX) and was used as received. Characterization. (a) Sorption. Propane solubility in PDMS was determined at different pressures and

temperatures using using a high-pressure barometric apparatus.30 Initially, a polymer film was placed in the sample chamber and exposed to vacuum overnight to remove air gases. A known amount of penetrant gas was introduced into the chamber, and the pressure was allowed to equilibrate. Once the chamber pressure was constant, the amount of gas sorbed by the polymer was determined by mass balance. Additional penetrant was then introduced, and the procedure was repeated. In this incremental manner, penetrant uptake was determined as a function of pressure. The maximum pressure was 3.4-8.3 atm depending on the temperature. Sorption equilibrium for all gases was reached within, at most, a few hours. After each isotherm was measured, the polymer sample was degassed under vacuum overnight. The system temperature was controlled to (0.1 °C using a constant-temperature bath. The sorption experiments were performed in the following temperature order: 35, 55, 20, and 0 °C. (b) Permeation. Pure propane permeability at a permeate (downstream) pressure of 1 atm was determined in PDMS using a constant-pressure/variablevolume apparatus.31 The membrane area was 13.8 cm2. The upstream pressure was varied from a minimum of 1.1-1.7 atm to a maximum of 1.95-8.5 atm, depending on the temperature. Gas flow rates were measured with a soap-film bubble flowmeter. The system temperature was controlled to (0.5 °C using a constant-temperature bath. Prior to each experiment, the upstream and downstream sides of the permeation cell were purged with penetrant gas. The permeation experiments were conducted in the order of decreasing temperature from 55 to -20 °C. When steady-state conditions were attained, the following expression was used to evaluate the permeability, P, (cm3(STP) cm/(cm2 s cmHg)):

P)

(

)(

)( )( )

p1 dV 22,414 l A p2 - p1 RT dt

(24)

where p2 is the upstream pressure (cmHg), p1 is the downstream pressure (atmospheric pressure (cmHg) in this case), l is the membrane thickness (cm), A is the membrane area (cm2), T is the absolute temperature (K), R is the universal gas constant (6236.56 cm3 cmHg/(mol K)), and dV/dt is the volumetric displacement rate of the soap film in the bubble flowmeter (cm3/s). The permeability coefficient P is often expressed in units of Barrers, where 1 Barrer equals 10-10 cm3(STP) cm/(cm2 s cmHg). Pure propane permeability at a permeate pressure of 0 atm was measured in a constant-volume/variablepressure apparatus.32 The membrane area was 13.8 cm2. The upstream pressure was varied from 1.3 to 2.3 atm. The downstream side was maintained below 10 mmHg. Prior to each experiment, the upstream and downstream sides of the permeation cell were evacuated to below 0.5 mmHg. The system temperature was controlled to (0.5 °C using a constant-temperature bath. The increase in pressure on the downstream side was recorded using a data acquisition system employing LABTECH software. When the rate of pressure increase on the downstream side, dp/dt (cmHg/s), attained its pseudo-steady-state value, the permeability P (cm3(STP) cm/(cm2 s cmHg)) was calculated using the expression

P)

(

)( )( )( )

22,414 l V dp A pabs RT dt

(25)

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Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 Table 2. Model Parameters for Solubility Data adjustable constants in eq 23 penetrant

polymer

propane halothane methyl bromide isobutylene n-hexane

PDMS PDMS PE PE PE

χ0

χ1 (K)

error in model prediction (%)a χ2

av std dev max

-0.64 518 -0.97 1.9 0.52 3.6 -5.97 863 5.11 1.3 -2.38 1159 1.2 -0.63 603 2.3

1.6 2.4 1.1 1.2 1.7

8.9 8.6 3.2 3.5 4.5

a Percentage error in model prediction ) |(C model - Cexpt)/Cexpt| × 100. The magnitude and variation of prediction errors in individual experimental points are characterized by the average, maximum, and standard deviation of these error values.

Table 3. Model Parameters for Permeability Data adjustable constants in eqs 19 and 20

Figure 3. Sorption isotherms of propane in PDMS at 0-55 °C. The lines represent Flory-Huggins fits to the experimental data and are based on the adjustable constants in Table 2. penetrant

polymer

EDo (kJ/mol)

k (10-3 cm3/ cm3(STP))

b

propane halothane methyl bromide isobutylene n-hexane

PDMS PDMS PE PE PE

11.3 15 55 53 59

5.35 0.10 7.45 19.5 14.85

10.2 10.6 7.6 8.6 7.8

of 8,580 Barrers at 35 °C in the limit of negligible pressure drop across the membrane,34 which is in reasonable agreement with the value of about 6,500 Barrers obtained in this study under the same temperature and pressure conditions. Model Fitting Procedure

Figure 4. Propane permeability in PDMS as a function of upstream pressure at -20 to 55 °C. Downstream pressure was 1 atm. The lines represent model fits to the experimental data and are based on the adjustable constants in Table 3.

where pabs is the upstream pressure (cmHg) and V is the downstream volume (cm3). Experimental Results Figure 3 displays sorption isotherms of propane in PDMS from 0 to 55 °C. The isotherms are convex to the pressure axis, which is typical for sorption of condensable penetrants in rubbery polymers.3 The curvature of the isotherms decreases with increasing temperature, suggesting a weaker dependence of solubility on pressure at higher temperatures. This is consistent with the findings of Shah et al.,33 who observed a decrease in the pressure dependence of propane solubility in PDMS with increasing temperature. Shah et al. report infinite dilution propane solubilities of 6.45 and 4.04 cm3(STP)/ (cm3 atm) at 35 and 55 °C, respectively. Our values (6.5 and 4.2 cm3(STP)/(cm3 atm), respectively) are in excellent agreement with theirs. Figure 4 displays the propane permeability coefficient for propane in PDMS as a function of upstream pressure over the temperature range of -20 to 55 °C. The downstream pressure was 1 atm. Propane permeability increases with decreasing temperature at any given pressure. This is consistent with previous observations of the temperature dependence of propane permeability in PDMS.34 Stern et al. obtained a propane permeability

As indicated in Theory, a sorption model is required to calculate penetrant concentration values for fitting the proposed permeability model (eq 19). Because all model cases in this study correspond to condensable penetrant transport in rubbery polymers, the FloryHuggins equation was used to describe penetrant sorption. Equation 23 was used to describe the temperature and concentration dependences of the χ parameter in the Flory-Huggins equation. The values of the adjustable constants in eq 23 and the quality of the fits are reported in Table 2. The model fits, shown in Figure 3 as smooth curves through the data points for propane sorption in PDMS, are in excellent agreement with the experimental data. The permeability model (eq 19) contains two adjustable constants, EDo and k. In addition, b is treated as an adjustable constant, despite its best-fit value determined from Figure 2, due to the scatter in the data around the best-fit linear trend line in this figure. Bestfit values of these parameters were determined by a nonlinear least-squares fit to experimental permeability data over a range of pressures and temperatures, and they are listed in Table 3 for all penetrant-polymer pairs in this study. The smooth curves in Figure 4 represent model fits to the propane-PDMS data using the parameters in Table 3. The fits are, in most cases, within the experimental uncertainty of the data. Results and Discussion Propane in PDMS. From Figure 3 and Table 2, the Flory-Huggins equation provides an excellent description of propane sorption in PDMS with a concentrationand temperature-dependent χ parameter. Penetrant concentration values from this equation and experimen-

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Figure 5. Quality of new model fits to experimental permeability data for condensable penetrants in (a,b) PDMS and (c,d,e) PE.

Figure 6. Halothane sorption isotherms in PDMS at 21-50 °C.9 The lines represent Flory-Huggins fits to the experimental data and are based on the adjustable constants in Table 2.

Figure 7. Permeability of halothane in PDMS at 17-60 °C.9 The lines represent model fits to the experimental data and are based on the adjustable constants in Table 3.

tal permeability data from Figure 4 were used to fit the permeability model in eq 19. The best-fit values of the adjustable constants, EDo, k, and b, are listed in Table 3. The quality of the resulting fit can be judged from Figures 4 and 5a. From Table 3, the best-fit value for EDo is 11.3 kJ/mol. This number compares well with the ED value of 11.8 kJ/mol reported by Stern et al., based on diffusivity values in the infinite dilution limit over the temperature range of 10-55 °C.34 The best-fit value of b obtained from the fitting procedure with the current data set is 10.2, which is higher than the value of 8.3 given by the best-fit trend line in Figure 2. However, using our EDo and b values in eq 9, the calculated infinite-dilution Do value is 5.6 × 10-4 cm2/s, which is close to the value of 9 × 10-4 cm2/s reported by Stern et al.34 Halothane in PDMS. Halothane (CF3CHClBr) solubility and permeability data in PDMS are presented in Figures 6 and 7, respectively.9 Halothane sorption in PDMS is adequately modeled by the Flory-Huggins equation with a constant χ parameter (Table 2 and

Figure 6). The best-fit values of the adjustable constants of eq 19 for this penetrant-polymer pair are reported in Table 3. With these parameters, the permeability data can be described with less than 10% error (except for one point), as shown in Figure 5b, which is probably the limit of the experimental measurements. The bestfit EDo value is 15 kJ/mol, which is in excellent agreement with the ED value of 14.8 kJ/mol calculated by Suwandi and Stern9 from the same data set, in the infinite dilution limit, by using eq 8. Similar to the case of propane transport in PDMS, the b value for this penetrant in PDMS is 10.6, which is higher than the best-fit value of 8.3 estimated from the data in Figure 2. Thus, PDMS seems to obey the linear free energy relationship with a b value that is higher than the values reported for other polymers. This can also be seen from Figure 2, where the points depicting penetrant transport in PDMS (solid symbols) are seen to lie at the outer fringes of the data scatter in the figure. The fundamental basis for this discrepancy is not known.

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Figure 8. Sorption isotherms of (a) methyl bromide, (b) isobutylene, and (c) n-hexane in PE.8 The lines represent FloryHuggins fits to the experimental data and are based on the adjustable constants in Table 2.

Various Organic Vapors in Poly(ethylene). Methyl bromide, isobutylene, and n-hexane sorption isotherms in poly(ethylene) are presented in parts a-c of Figure 8.8 Sorption of n-hexane and isobutylene in polyethylene is very well described by the FloryHuggins model with a temperature-dependent χ parameter, while methyl bromide sorption requires a concentration- and temperature-dependent χ parameter (cf. Table 2). The permeability coefficients of these three condensable penetrants in PE are shown in parts a-c of Figure 9.8 Equation 19 can predict permeabilities for the three penetrants in PE with less than 10% error for practically all of the data (cf. Figures 5c-e), with the best-fit values reported in Table 3. From this table,

Figure 9. Permeabilities of (a) methyl bromide, (b) isobutylene, and (c) n-hexane in PE at -8 to 30 °C.8 The lines represent model fits to the experimental data and are based on the adjustable constants in Table 3.

b values for the three penetrants in PE are much closer to the best-fit value of 8.3 (from Figure 2) than the b values obtained for condensable penetrants in PDMS. ED values for these penetrant-PE pairs could not be found in the literature for comparison. However, diffusion coefficients of penetrants in polymers typically decrease with increasing penetrant size.35 Diffusion coefficients often scale with penetrant critical volume, Vc, as35

D)

τ Vcη

(26)

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Figure 10. Correlation of the activation energy of diffusion ED of penetrants in PE with penetrant critical volume. The unfilled symbols are literature data,28 and the filled symbols are EDo values calculated from the new model for methyl bromide, isobutylene, and n-hexane. The solid line is fitted to all data points in the plot and has the equation ED (kJ/mol) ) 36.4 log(Vc (cm3/mol)) - 30.2.

Figure 11. Effect of permeate pressure on propane permeability in PDMS at -10 °C. The solid lines depict the model prediction based on best-fit values from Table 3. The open symbols are experimentally measured permeability values at a downstream pressure of 1 atm (O) and 0 atm (0). These experimental data were not used in determining the best-fit values of the model.

where τ and η are adjustable constants. From eqs 8 and 26, ED is expected to have a logarithmic dependence on Vc. Figure 10 presents ED values reported in literature for several penetrants in PE28 as well as the EDo values determined from the new model for the three penetrants in this PE study. A logarithmic trend line provides a good correlation between ED and Vc for these penetrants in PE. The above examples show that the proposed model (eq 19) can describe concentration- and temperaturedependent permeability data well with very few fitting parameters. Furthermore, values of the two parameters EDo and b, can be estimated from their values for other penetrants in the same polymer and by utilizing correlations between the activation energy of diffusion and penetrant size. These estimates can then be used to provide rough values of permeabilities, as a first approximation, in the absence of any experimental permeability information. Effect of Permeate Pressure on Permeability. For penetrants having a concentration-dependent permeability, variation in both feed and permeate pressures can change the permeability because both pressures affect penetrant concentration in the polymer. For example, Figure 11 shows predictions (solid lines) of the new model (eq 19) for propane permeability in PDMS at -10 °C and two different permeate pressures, using the best-fit values in Table 3. (No permeability data at -10 °C were used to determine these best-fit values.) The model predicts that permeability decreases as downstream pressure decreases. For example, at an upstream pressure of 2.36 atm, propane permeability is predicted to decrease by 24% when the downstream pressure is lowered from 1 to 0 atm. This prediction is of interest because decreasing the downstream pressure at fixed upstream pressure may be used to increase the driving force for permeation and, hence, gas flux through a membrane. In the current example, the driving force is increased by 73.5% as downstream pressure decreases from 1 atm to vacuum. The standard model, as given by eq 1, would predict a constant permeability coefficient with a change in permeate pressure. Thus, eq 2 would predict a 73.5% increase in flux. However, both experimentally and according to the new model, flux increases by only 32% because of the concomitant decrease in

permeability with decreasing permeate pressure. Therefore, the new model is able to predict the effect of permeate pressure on permeability coefficients, thus providing more reliable permeability values for design calculations. Conclusions Propane solubility in PDMS increases with decreasing temperature, and the sorption isotherms are welldescribed by the Flory-Huggins equation with a concentration- and temperature-dependent χ parameter. Propane permeability in PDMS increases with decreasing temperature and is well-described by the model presented in this paper. Propane permeability decreases with decreasing permeate pressure, and both the magnitude and direction of this permeability change are captured by the new model. The model requires few adjustable constants and may be useful as a first step to provide a rational framework for estimating permeability coefficients in rubbery polymers under operating conditions that are not in the range of those used to acquire experimental data. Acknowledgment We gratefully acknowledge partial support of this work by the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy (Contract No. DE-FG03-02ER15362). Literature Cited (1) Baker, R. W. Membrane Technology and Applications; McGraw-Hill: New York, 2000. (2) Ohlrogge, K.; Stu¨rken, K. In Membrane Technology in the Chemical Industry; Nunes, S. P., Peinemann, K.-V., Eds.; WileyVCH: Weinheim, Germany, 2001; pp 71-94. (3) Ghosal, K.; Freeman, B. D. Gas Separation Using Polymer Membranes: An Overview. Polym. Adv. Technol. 1994, 5, 673697. (4) Park, G. S. Diffusion of Some Halomethanes in Polystyrene. Trans. Faraday Soc. 1950, 46, 684-697. (5) Prager, S.; Long, F. A. Diffusion of Hydrocarbons in Polyisobutylene. J. Am. Chem. Soc. 1951, 73, 4072-4075. (6) Park, G. S. The Diffusion of Some Organic Substances in Polystyrene. Trans. Faraday Soc. 1951, 47, 1007-1013.

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Received for review August 5, 2004 Revised manuscript received November 10, 2004 Accepted November 18, 2004 IE0492909